Mr Cameron will tender his resignation to the Queen on Wednesday, leaving Mrs May, and home secretary since 2010, to appoint her own ministerial team.
Mrs May said she was “honoured and humbled” to be taking over and pledged to make a success of the UK’s EU exit.
David Cameron will chair his final cabinet meeting as prime minister as Theresa May prepares to take over.
Mrs May had been expecting a nine-week Conservative leadership race, but rival Andrea Leadsom withdrew on Monday.